
Book Nr. 6 Critical Properties of Phi^4Theories
and how to extract the critical properties of the theory from the resulting divergent power series. These properties are calculated for various secondorder phase transitions of threedimensional systems with high accuracy, in particular the critical exponents observable in experiments close to the phase transition. Beginning with an introduction to critical phenomena, this book develops the functionalintegral description of quantum field theories, their perturbation expansions, and a method for finding recursively all Feynman diagrams to any order in the coupling strength. Algebraic computer programs are supplied on accompanying World Wide Web pages. The diagrams correspond to integrals in momentum space. They are evaluated in 4e dimensions, where they possess pole terms in 1/e. The pole terms are collected into renormalization constants. The theory of the renormalization group is used to find the critical scaling laws. They contain critical exponents which are obtained from the renormalization constants in the form of power series. These are divergent, due to factorially growing expansion coefficients. The evaluation requires resummation procedures, which are performed in two ways:
of the specific heat of superfluid helium is shown to agree very well with the extremely precise experimental number found in the space shuttle orbiting the earth (whose data are displayed on the cover of the book). The phi4theories investigated in this book contain any number N of fields in an O(N)symmetric interaction, or in an interaction in which O(N)symmetry is broken by a term of a cubic symmetry. The crossover behavior between the different symmetries is investigated. In addition, alternative ways of obtaining critical exponents of phi4theories are sketched, such as variational perturbation expansions in three rather than 4e dimensions, and improved ratio tests in hightemperature expansions of lattice models. 

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